Question

Graph each function.

$f\left(x\right)=\sqrt{16-4x^2}$.

Short answer

The graph of the function is half an ellipse.

Step-by-step solution

Step 1. Given information.

We have given function $y=\sqrt{16-4x^2}$

Now, find equation of conic in a standard form we have

role="math" localid="1646757247233" $$\begin{gathered} y^2={\left(\sqrt{16-4x^2}\right)}^2 \\ {y}^2=16-4x^2 \\ {y}^2+4x^2=16 \\ \frac{y^2}{16}+\frac{4x^2}{16}=\frac{16}{16} \\ \frac{y^2}{16}+\frac{x^2}{4}=1 \\ \frac{x^2}{2^2}+\frac{y^2}{4^2}=1 \end{gathered}$$

Step 2. Graph of the function.

Substitute $b=2$and $a=4$in$b^2=a^2-c^2$.

$$\begin{gathered} c^2=4^2-2^2 \\ {c}^2=12 \\ c=\sqrt{12} \\ c=2\sqrt{3} \end{gathered}$$

The foci at $(0,\pm 2\sqrt{3})$.

The vertices of the major axis of the ellipse are at $(0,4)$and $(0,-4)$.

The vertices of the minor axis of the ellipse are at $(2,0)$and $(-2,0)$.

Use the vertices and foci of the ellipse to graph the function is